x-sin-x-1-cos-x-dx-

Question Number 112847 by bemath last updated on 10/Sep/20

\int \frac{x - \sin x}{1 - \cos x} dx ?

Answered by bobhans last updated on 10/Sep/20

I = \int \frac{x - \sin x}{1 - \cos x} dx = \int \frac{x - 2 \sin (\frac{x}{2}) \cos (\frac{x}{2})}{2 \sin^{2} (\frac{x}{2})} dx

I = \frac{1}{2} \int x cosec^{2} (\frac{x}{2}) dx - \int \cot (\frac{x}{2}) dx

I_{1} = \frac{1}{2} \int x cosec^{2} (\frac{x}{2}) dx = - \int x d (\cot (\frac{x}{2}))

= - (x . \cot (\frac{x}{2})) + \int \cot (\frac{x}{2}) dx

I = I_{1} - \int \cot (\frac{x}{2}) dx

I = - xcot (\frac{x}{2}) + \int \cot (\frac{x}{2}) dx - \int \cot (\frac{x}{2}) dx

I = - x \cot (\frac{x}{2}) + c

Commented by bemath last updated on 10/Sep/20

santuuuyyy

Facebook Tweet Pin